Optimal Packings of Two to Four Equal Circles on Any Flat Torus

TL;DR

This paper derives explicit formulas for the densest arrangements of two to four equal circles on any flat torus, proving their optimality with advanced mathematical techniques.

Contribution

It provides the first explicit formulas for optimal circle packings of small numbers of equal circles on any flat torus, along with rigorous proofs of their optimality.

Findings

Explicit formulas for circle radii and positions on flat tori.

Proofs confirming the optimality of these arrangements.

Applicable to all flat tori generated by two vectors.

Abstract

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.